Question

The letters of the word 'LOGARITHM' are arranged at random. The probability that the arrangement starts with a vowel and ends with a consonant is

• A. \( \frac{7!}{9!} \)
• B. \( \frac{18}{9!} \)
• C. \( \frac{1}{4} \)
• D. \( \frac{1}{9} \)

Hint:

When calculating the probability of arrangements, remember to account for the fixed positions of specific characters and calculate based on the remaining arrangements.

Answer 1

The Correct Option is C

Step 1: Identifying vowels and consonants.
In the word "LOGARITHM," the vowels are O, A, I. The consonants are L, G, R, T, H, M. There are 3 vowels and 6 consonants.

Step 2: Calculating the probability.
We are interested in arrangements where the word starts with a vowel and ends with a consonant. The number of ways to arrange the vowels in the first position and consonants in the last is \( 3 \times 6 \), and the number of ways to arrange the remaining letters is \( 6! \).
Thus, the total favorable outcomes are: \[ 3 \times 6 \times 6! \] The total number of possible arrangements is \( 8! \). Therefore, the probability is: \[ \frac{3 \times 6 \times 6!}{8!} = \frac{1}{4} \]
Step 3: Conclusion.
Thus, the probability is \( \frac{1}{4} \), which makes option (C) the correct answer.